Labellings for assumption-based and abstract argumentation

The semantics of Assumption-Based Argumentation (ABA) frameworks are traditionally characterised as assumption extensions, i.e. sets of accepted assumptions. Assumption labellings are an alternative way to express the semantics of flat ABA frameworks, where one of the labels in, out, or undec is assigned to each assumption. They are beneficial for applications where it is important to distinguish not only between accepted and non-accepted assumptions, but further divide the non-accepted assumptions into those which are clearly rejected and those which are neither accepted nor rejected and thus undecided. We prove one-to-one correspondences between assumption labellings and extensions for the admissible, grounded, complete, preferred, ideal, semi-stable and stable semantics. We also show how the definition of assumption labellings for flat ABA frameworks can be extended to assumption labellings for any (flat and non-flat) ABA framework, enabling reasoning with a wider range of scenarios. Since flat ABA frameworks are structured instances of Argumentation (AA) frameworks, we furthermore investigate the relation between assumption labellings for flat ABA frameworks and argument labellings for AA frameworks. Building upon prior work on complete assumption and argument labellings, we prove one-to-one correspondences between grounded, preferred, ideal, and stable assumption and argument labellings, and a one-to-many correspondence between admissible assumption and argument labellings. Inspired by the notion of admissible assumption labellings we introduce committed admissible argument labellings for AA frameworks, which correspond more closely to admissible assumption labellings of ABA frameworks than admissible argument labellings do.